
Model I: Ref. [1]} proposed a simple anomaly-free theory for the spontaneous breaking of local baryon number where the anomalies
are canceled with the fields listed in Table REF .
<TABLE>The relevant Lagrangian for our discussion is given by
\({\cal {L}} &\supset & y_1 \bar{\Psi }_L H \eta _R + y_2 \bar{\Psi }_L \tilde{H} \chi _R + y_3 \bar{\Psi }_R H \eta _L + y_4 \bar{\Psi }_R \tilde{H} \chi _L \nonumber \\&+& \lambda _1 \bar{\Psi }_L \Psi _R S_B^* + \lambda _2 \bar{\eta }_R \eta _L S_B^* + \lambda _3 \bar{\chi }_R \chi _L S_B^* + \frac{\lambda _\chi }{\sqrt{2}} \chi _L \chi _L S_B^* + \lambda _5 \chi _R \chi _R S_B + \rm {h.c.},\) 
where \(H \sim (1,2,1/2, 0)\)  and \(S_B \sim (1,1,0,3)\) . In general, the DM candidate can be a linear combination of all neutral fields
and it is a Majorana fermion. Recently, this model has been investigated in Ref. [2]}. See also a recent
study in Ref. [3]} where one can have a similar model where the DM candidate is a Dirac fermion.


Model II: In the model proposed in Ref. [4]} the anomalies are canceled with only four extra fields listed in Table REF .
<TABLE>The Yukawa interactions needed to generate vector-like masses for the new fermions are given by
\({\cal {L}} &\supset & y_1 \bar{\Psi }_R H \chi _L + y_2 H^\dagger \Psi _L \chi _L + y_3 H^\dagger \Sigma _L \Psi _L + y_4 \bar{\Psi }_R \Sigma _L H \nonumber \\&+& \lambda _1 \bar{\Psi }_L \Psi _R S_B + \frac{\lambda _\chi }{\sqrt{2}} \chi _L \chi _L S_B + \lambda _\Sigma \text{Tr} \Sigma _L^2 S_B + \rm {h.c.},\) 
the DM is always a Majorana fermion and it can be a linear combination of all neutral fields present in the theory.
See Ref. [5]} for a detailed phenomenological study of this model.

